Question

what is the measure of ∠c to the nearest whole degree?
70°
77°
80°
85°
law of cosines: a² = b² + c² - 2bccos(a)

Explanation:

Step1: Identify the sides for cosine - law

In right - triangle \(ABC\) with right - angle at \(B\), \(a = 2\), \(b=\sqrt{8.17}\), \(c = 9\). We can use the law of cosines \(\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}\). First, find \(b^{2}=8.17\), \(a^{2}=4\), \(c^{2}=81\).

Step2: Substitute values into the formula

\(\cos C=\frac{4 + 8.17-81}{2\times2\times\sqrt{8.17}}=\frac{12.17 - 81}{4\sqrt{8.17}}=\frac{- 68.83}{4\sqrt{8.17}}\).
\(\sqrt{8.17}\approx2.86\), so \(4\sqrt{8.17}\approx4\times2.86 = 11.44\). Then \(\cos C=\frac{-68.83}{11.44}\approx - 6\) (this is wrong. Let's use the right - triangle trigonometry).
Since \(\tan C=\frac{AB}{BC}\), and \(AB=\sqrt{AC^{2}-BC^{2}}=\sqrt{9^{2}-2^{2}}=\sqrt{81 - 4}=\sqrt{77}\approx8.77\), \(\tan C=\frac{\sqrt{77}}{2}\approx\frac{8.77}{2}=4.385\).

Step3: Find the angle

\(C=\arctan(4.385)\). Using a calculator, \(C=\arctan(4.385)\approx77^{\circ}\).

Answer:

B. \(77^{\circ}\)